I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mathbb C$-valued functions on the upper half plane (Chapter 2.1):
I was not able to verify (1.4) by the way. Maybe I made a mistake, but from what I calculated, the first equality was not correct. I was wondering whether one of the definitions (1.1), (1.2), (1.3) was not written correctly.
Edit: I have checked again, and (1.4) is actually correct.
Wikipedia defines the weight $k$ non-Euclidean Laplacian by
$$\Delta_k = -y^2( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) + iky (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y})$$
Which definition is correct?